On functors from category of Giry algebras to category of convex spaces

TL;DR

This paper refutes the claim that the category of convex spaces is equivalent to the category of Giry monad algebras, providing a detailed analysis and counterexamples to previous assertions.

Contribution

It demonstrates that the previously claimed equivalence between convex spaces and Giry monad algebras does not hold, clarifying the structure of these categories.

Findings

The category of convex spaces is not equivalent to Giry monad algebras.

Counterexamples disprove previous assertions of equivalence.

Parallel results are shown for super convex spaces.

Abstract

In The factorization of the Giry monad (arXiv:1707.00488v2) the author asserts that the category of convex spaces is equivalent to the category of Eilenberg-Moore algebras over the Giry monad. Some of the statements employed in the proof of this claim have been refuted in our earlier paper (arXiv:1803.07956). Building on the results of that paper we prove that no such equivalence exists and a parallel statement is proved for the category of super convex spaces.